Examples
- Every compact space is paracompact.
- Every regular Lindelöf space is paracompact. In particular, every locally compact Hausdorff second-countable space is paracompact.
- The Sorgenfrey line is paracompact, even though it is neither compact, locally compact, second countable, nor metrizable.
- Every CW complex is paracompact
- (Theorem of A. H. Stone) Every metric space is paracompact. Early proofs were somewhat involved, but an elementary one was found by M. E. Rudin. Existing proofs of this require the axiom of choice for the non-separable case. It has been shown that neither ZF theory nor ZF theory with the axiom of dependent choice is sufficient.
Some examples of spaces that are not paracompact include:
- The most famous counterexample is the long line, which is a nonparacompact topological manifold. (The long line is locally compact, but not second countable.)
- Another counterexample is a product of uncountably many copies of an infinite discrete space. Any infinite set carrying the particular point topology is not paracompact; in fact it is not even metacompact.
- The Prüfer manifold is a non-paracompact surface.
Read more about this topic: Paracompact Space
Famous quotes containing the word examples:
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)